We successfully realized a 1.5µm optical frequency-tunable Cs atomic clock by employing a mode-hop-free 9.19GHz mode-locked erbium fiber laser with an optical etalon in the cavity. We achieved independent control of the optical frequency and the repetition rate (microwave frequency) of the 9.19GHz optical pulse, in which the microwave frequency was controlled without disturbing the optical frequency. The optical frequency was continuously tuned over 1GHz without changing the repetition rate, indicating the possibility of the absolute frequency stabilization of the Cs optical atomic clock.